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Proving a Premise with Z Ratio

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Inferential statistics serve as the basis for much of the scientific thinking and reasoning that you have been reading about in this text. This is especially true with regard to proving a premise before including it within an argument. Inferential statistics involves drawing samples from populations, quantifying the data pertinent to each participant or phenomenon, and then making inferences about the entire population through the examination of the data contained within a sample. One such technique involves drawing two samples from their respective populations, computing the means and standard deviations relative to a designated quantitative or continuous variable (such as age, or education) and then testing to see if there is a statistically significant difference between the means of the two samples. Proving a Premise with Z ratio

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If the difference is so significant that it could not have happened by mere chance, then the inference is made that there is a significant difference between the means of not only the sample, but probably the total population as well. In other words, we "infer" the findings of the sample to the total population, based on probability. A critical ratio test (Z ratio) can be used to determine if the difference between the means is statistically significant or not. As applied to critical thinking, before we state in an argument that age is a factor in involvement in driver safety and involvement in traffic collisions, we should test the hypothesis by collecting two samples at random, one for a group of people who have been involved in a collision over the past three years and another group of people who have not been involved in a traffic collision. Proving a Premise with Z ratio

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If we find that the mean age for the accident group was 22 and the mean age for the non-accident group was 32, we would see a visible difference between the mean ages. If the Z ratio was computer beyond a level needed for statistically significant difference (say 2.65), then we could conclude safely that the same level of age disparity observed in the sample probably exists in the total population as well and thus our assertion that collision participants tend to be younger would be a safe presumption Conversely, if we draw two samples with the same mean age, and test the hypothesis to discover there is no statistically significant difference in the mean number of traffic collisions, we have furthered our understanding that [age] might play a role in determining propensity for safe driving. Again, such a comparative analysis does not prescribed [why] this occurs, only that there is a measurable and statistically significant result. Proving a Premise with Z ratio

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Statistical significance (based on the laws of probability) is a conventional way of stating whether a difference between groups or a relationship between variables has occurred simply by chance, or not. When comparing means of two samples, a Z ratio helps the researcher decide whether the difference would be expected to occur by chance, or whether it would not be expected to happen by chance. This also facilitates our determination of whether the difference may be attributed to random sampling error, or whether the difference between the means was so large that it overcame expected sampling error. The.05 and.01 levels have been accepted by the scientific community as the standard cutoff points of acceptability for such measurement. Proving a Premise with Z ratio

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Think of it this way, our goal is to measure whether the difference is so significant that it could not have happened by mere chance. To accomplish this, we have to rule out that the difference between the means we found in our samples was caused by mere chance or sampling error. Accordingly, we need to conduct an empirical test to assure ourselves that the premise (that age is a contributive factor) is correct. We accomplish this by incorporating a quantitative analysis that examines the means of two samples to see if there is a [significant] difference between the mean ages of people involved in traffic accidents. To be absolutely sure that sampling error wasn't the cause of the difference that we noted between the two groups, we start out by accepting the null hypothesis (which would state; There is NO statistically significant difference between the mean age for those who are involved in traffic accidents and those who have not experienced such problems) until the Z Ratio proves, beyond reasonable doubt or mathematic certainty, that such a difference exists. Proving a Premise with Z ratio

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The.05 and.01 levels of significance correspond to the 95% and 99% (or plus 2 and plus 3) standard deviations on a normal distribution bell curve. If a difference between means produces a Z ratio that is large enough (plus or minus 1.96) so that it would be expected to occur by chance in less than 5% of the cases (.05 or 5 times out of 100), then the difference between means is said to be significant at the 5% or (.05) level. If we see a Z ratio equal to or larger than 1.96, on either side of the curve, we can safely "reject" the null hypothesis and accept the research hypothesis, which says, "there IS a significant difference between the two groups relative to mean age", subsequently we can also safely presume that age is ONE of the factors that influences driver safety. Proving a Premise with Z ratio

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Z ratio is a statistical test that can be used to examine the difference between means of samples drawn from populations. It is considered a critical ratio test for sample sizes larger than 30 (N > 30). Z ratio assumes normality of the distribution. Areas under the normal curve may be examined to determine within that level of probability the difference observed betweens means would have occurred by chance. Proving a premise before including it within an argument is a requirement if you stand any chance at all of being accurate. To say that age matters in determining driver safety might be an interesting presumption, but certainty requires that the arguer take the time to establish the research protocol that specifically tests the accuracy of such a presumptive statement. Proving a Premise with Z ratio

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Proof is essential before forming a conclusion and creating policy. There may be (and usually are) many variables that influence the outcome of things. Age might be one of 50 variables that contribute to driver safety. The Z ratio lets you prove the truth of the premise, but does not disclose what the other remaining 49 variables might be in such an equation. To identify all 49 factors, you have to study the problem, identify potential influences, test each one of them individually to discern whether they have any contributive value, and from this aggregate analysis, you can formulate a comprehensive argument about the factors affecting driver safety. Proving a Premise with Z ratio

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Problem Statement: Rehabilitating Juvenile Offenders Research Question: What is the relationship between the recidivism rates for juvenile offenders who received treatment as part of a diversion program versus those offenders who were strictly confined in a youth detention facility? Research Hypothesis: There is a statistically significant difference in the mean re-arrest rate per client for those treated in the diversion program as compared to those who simply receive incarceration. Null Hypothesis: There is no statistically significant difference in the mean re-arrest rates between these two groups. Probability Statement: Upon initial arrest, juvenile offenders were randomly selected and either assigned to undergo diversion treatment as part of their sentence or they were placed exclusively within juvenile detention. Re-arrest records were maintained for a one year period for all juveniles included within the study. For a one tailed test, a Z ratio of 1.64 is needed for significance at the.05 level, and 2.33 for significance at the.01 level. Proving a Premise with Z ratio

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Presentation of the Data: Mean Re-Arrests and Standard Deviations Of Juvenile Offenders N M S Incarceration4001.2.90 Diversion400.75.25 Z Ratio = 9.63, P<.01 Examination of the Null Hypothesis: For this analysis a Z ratio of 9.63 was computed. The NH is rejected at the.01 level of significance, P<.01. Analysis and Interpretation: Predicated on the results of this analysis, it appears prudent to conclude that diversion programs do have a favorable impact in preventing recidivism rates among juvenile offenders. As applied to the samples, we see that for those juveniles who were simply incarcerated, the mean re-arrest rate was 1.2 arrests during the subsequent year, while for those who received diversion, the mean re-arrest rate was only.75 instances. Clearly diversion isnt the only factor associated with juvenile delinquency, but based on these data; treatment of offenders (as opposed to mere incarceration) does appear to have a positive effect in influencing future behavior. Proving a Premise with Z ratio

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In the next slide is an example of how we might structure a research design, using Z ratio, to test whether there is a statistically significant difference in the mean test scores for an examination administered to measure knowledge of the subject, between people who have completed formal instruction in critical thinking and those who have not completed such training. The underlying theoretical framework for such a contention would naturally be oriented toward an assertion that training improves ability. Proving a Premise with Z ratio

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Problem: Determining the effectiveness of critical thinking education in undergraduate programs. Research Question: What is the relationship between mandatory critical thinking and reasoning education at the undergraduate level and test scores measuring proficiency in this academic area? Research Hypothesis: There is a statistically significant difference in the mean test scores between students who were required to complete a class in critical thinking during their undergraduate education and students who did not complete such a class. Null Hypothesis: There is no significant difference between mean test scores for these two groups. Probability Statement: Prior to graduation, students were randomly selected to participate in an experiment that endeavored to measure their knowledge and familiarity with the concepts of critical thinking and reasoning. Students were grouped according to whether they were required to successfully complete a class in critical thinking during their undergraduate program versus those who did not complete such a course of instruction. For a two-tailed test, a Z ratio of +/- 1.96 is needed for significance at the.05 level and +/- 2.58 at the.01 significance level. Proving a Premise with Z ratio

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Means and Standard Deviations of Students Test Scores N M S Completed Critical Thinking 361 94 5 No Critical Thinking Class 324 83 3 Z Ratio = 35.31, P<.01 Examination of the Null Hypothesis: The NH is rejected at the.01 level of significance; p<.01 Interpretation: Analysis of the data indicate that there is a statistically significant difference between the two groups, measured at the.01 level, suggesting that college students who are required to complete a course in critical thinking during their undergraduate program score higher than other students given the same examination. These findings suggest that completion of a critical thinking class during the undergraduate years serves to produce a more highly refined level of reasoning in college students (as measured by the average scores on the examination) and it can be subsequently argued that the experience better prepares students to engage in reasoning and logic. Proving a Premise with Z ratio

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Problem: Determining the impact of clear cut forestry practices on soil erosion. Research Question: What is the relationship between clear cut forest harvesting versus selective cut forestry practices as applied to soil erosion? Research Hypothesis: There is a statistically significant difference in the average number of inches of soil lost to erosion between regions using clear cut harvesting practices versus regions where selective cutting is used. Null Hypothesis: There is no significant difference in soil erosion as measured in annualized loss as measured in inches between these two harvesting practices. Probability Statement: To facilitate this analysis, a representative sample of ten (separate) forest regions was used for each category of harvesting method. Data pertinent to soil erosion was collected for each type of region (cut clear and selective cut methods), and assembled. The data represents average soil erosion as measured in inches of top soil lost for the five preceding years. For a two-tailed test, a Z ratio of +/- 1.96 is needed for significance at the.05 level and +/- 2.58 at the.01 significance level. Proving a Premise with Z ratio

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Means and Standard Deviations of Soil Erosion N M S Clear Cut Regions 10 1.2.35 Selective Cut Regions 10.95.31 Z Ratio = 1.69 Examination of the Null Hypothesis: The NH is accepted at the.05 level of significance; p>05 Interpretation: Analysis of the data indicates that there is not a statistically significant difference between the two samples. Although the descriptive information clearly reflects that the mean soil loss rate for the clear cut regions was larger (1.2 inches per year) than for those areas where selective cutting was utilized (.95 inches per year), the difference computed was not so large that it could have happened by mere chance. Subsequently we must conclude that there is no statistically significant difference in the mean soil erosion rates for regions harvested using these two forestry practices. Proving a Premise with Z ratio

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© 2008 McGraw-Hill Higher Education The Statistical Imagination Chapter 9. Hypothesis Testing I: The Six Steps of Statistical Inference.

© 2008 McGraw-Hill Higher Education The Statistical Imagination Chapter 9. Hypothesis Testing I: The Six Steps of Statistical Inference.

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